Properties

Label 2-684-4.3-c2-0-48
Degree $2$
Conductor $684$
Sign $0.800 - 0.598i$
Analytic cond. $18.6376$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.895 + 1.78i)2-s + (−2.39 − 3.20i)4-s + 5.60·5-s + 0.452i·7-s + (7.87 − 1.41i)8-s + (−5.02 + 10.0i)10-s − 3.07i·11-s + 0.732·13-s + (−0.809 − 0.405i)14-s + (−4.52 + 15.3i)16-s + 14.0·17-s − 4.35i·19-s + (−13.4 − 17.9i)20-s + (5.49 + 2.75i)22-s + 11.6i·23-s + ⋯
L(s)  = 1  + (−0.447 + 0.894i)2-s + (−0.598 − 0.800i)4-s + 1.12·5-s + 0.0646i·7-s + (0.984 − 0.176i)8-s + (−0.502 + 1.00i)10-s − 0.279i·11-s + 0.0563·13-s + (−0.0578 − 0.0289i)14-s + (−0.282 + 0.959i)16-s + 0.826·17-s − 0.229i·19-s + (−0.671 − 0.898i)20-s + (0.249 + 0.125i)22-s + 0.504i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.800 - 0.598i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.800 - 0.598i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.736406188\)
\(L(\frac12)\) \(\approx\) \(1.736406188\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.895 - 1.78i)T \)
3 \( 1 \)
19 \( 1 + 4.35iT \)
good5 \( 1 - 5.60T + 25T^{2} \)
7 \( 1 - 0.452iT - 49T^{2} \)
11 \( 1 + 3.07iT - 121T^{2} \)
13 \( 1 - 0.732T + 169T^{2} \)
17 \( 1 - 14.0T + 289T^{2} \)
23 \( 1 - 11.6iT - 529T^{2} \)
29 \( 1 - 27.5T + 841T^{2} \)
31 \( 1 + 50.9iT - 961T^{2} \)
37 \( 1 - 0.938T + 1.36e3T^{2} \)
41 \( 1 - 23.6T + 1.68e3T^{2} \)
43 \( 1 + 38.3iT - 1.84e3T^{2} \)
47 \( 1 - 2.34iT - 2.20e3T^{2} \)
53 \( 1 - 90.2T + 2.80e3T^{2} \)
59 \( 1 - 2.26iT - 3.48e3T^{2} \)
61 \( 1 + 24.9T + 3.72e3T^{2} \)
67 \( 1 - 23.0iT - 4.48e3T^{2} \)
71 \( 1 - 90.2iT - 5.04e3T^{2} \)
73 \( 1 - 102.T + 5.32e3T^{2} \)
79 \( 1 - 25.9iT - 6.24e3T^{2} \)
83 \( 1 - 90.2iT - 6.88e3T^{2} \)
89 \( 1 + 29.7T + 7.92e3T^{2} \)
97 \( 1 - 60.0T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.03294848748683451157370817012, −9.511494486925121020974107429052, −8.649966092975175143294004719088, −7.74636881906841428582596863160, −6.80013740808917245312463368512, −5.83350086479464304498593099167, −5.39757563935820587474077836740, −4.04349923752114921757367202980, −2.33479047063653263107809760965, −0.940591549469574194120409256946, 1.08865940129225707575372961140, 2.20220927272100795707246274466, 3.26325687264992171362619608557, 4.55801975692686068990958639937, 5.56224024344577939165160290309, 6.70389375996865221945979167265, 7.79222154614126610582466949777, 8.734975835530307054574168175747, 9.501133173092401106297198770683, 10.25963498995453342374682639192

Graph of the $Z$-function along the critical line